Homework SourseConsider a linear model Yi 0 1xi ei for i 1n where eis ar
Consider a linear model : Yi = 0 +1xi +ei for i = 1,...,n, where eis are iid normal random variables with mean = 0 and variance = ^2.
1. Find an expression for the least square estimator (ˆ) of for the model
Solution
Suppose we assume that there is a linear relationship between the Variables Y with variable X and disturbance term e then the model is Y=X + e and we are given with n observations or variables then the given observations may be written as
Y = 0 + 1Xi + ei for all i=1,2,……….n
The assumption of the model are
= if ij
Let us consider the model Y = 0 + 1Xi + ei
Where c 0 1 are parameters
ei = Y – 0 - 1Xi
(ei)2 = ( Yi - 0 - 1Xi )2
The above sum of squares is minimum of 0 1 for which the first derivative with respect 0 and 1 is equal to 0 the main principle for least square estimator is to minimize the sum of the squares of the deviations with respect to the parameters given
(ei)2 / 0 = 0
(ei)2 / 1 = 0
Consider (ei)2 / 0 = 0
/ 0 (( Yi - 0 - 1Xi )2) = 0
2(( Yi - 0 - 1Xi )(-1)) = 0
( Yi - 0 - 1Xi ) =0
Yi - n0 - 1 Xi = 0
n0 =Yi - 1 Xi
0 =Yi/n - 1 Xi /n
0 = Y - 1 X
(ei)2 / 0 = 0
(ei)2 / 1 = 0
Consider (ei)2 / 1 = 0
/ 1 (( Yi - 0 - 1Xi )2) = 0
2(( Yi - 0 - 1Xi )(- Xi)) = 0
( Xi Yi - 0 Xi - 1Xi2 ) =0
Xi Yi - 0 Xi - 1 Xi2 = 0
1 =Xi Yi - Xi Y / Xi2 - nX²
